Birkhoff integral for multi-valued functions

Abstract

The aim of this paper is to study Birkhoff integrability for multi-valued maps F :Ω→ cwk(X) , where (Ω,Σ,μ) is a complete finite measure space, X is a Banach space and cwk( X) is the family of all non-empty convex weakly compact subsets of X. It is shown that the Birkhoff integral of F can be computed as the limit for the Hausdorff distance in cwk( X) of a net of Riemann sums ∑ n μ( A n ) F( t n ). We link Birkhoff integrability with Debreu integrability, a notion introduced to replace sums associated to correspondences when studying certain models in Mathematical Economics. We show that each Debreu integrable multi-valued function is Birkhoff integrable and that each Birkhoff integrable multi-valued function is Pettis integrable. The three previous notions coincide for finite dimensional Banach spaces and they are different even for bounded multi-valued functions when X is infinite dimensional and X ∗ is assumed to be separable. We show that when F takes values in the family of all non-empty convex norm compact sets of a separable Banach space X, then F is Pettis integrable if, and only if, F is Birkhoff integrable; in particular, these Pettis integrable F's can be seen as single-valued Pettis integrable functions with values in some other adequate Banach space. Incidentally, to handle some of the constructions needed we prove that if X is an Asplund Banach space, then cwk( X) is separable for the Hausdorff distance if, and only if, X is finite dimensional.